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<h1 class="heading"><a href="linear-algebra-for-team-based-inquiry-learning.html"><span class="title">Linear Algebra for Team-Based Inquiry Learning</span></a></h1>
<p class="byline">Steven Clontz, Drew Lewis</p>
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<main class="main"><div id="content" class="pretext-content"><section xmlns:svg="http://www.w3.org/2000/svg" class="section" id="V1"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">2.1</span> <span class="title">Vector Spaces (V1)</span>
</h2>
<article class="observation remark-like" id="observation-3"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">2.1.1</span><span class="period">.</span>
</h6>
<p id="p-108">Several properties of the real numbers, such as commutivity:</p>
<div class="displaymath">
\begin{equation*}
x + y = y + x
\end{equation*}
</div>
<p data-braille="continuation">also hold for Euclidean vectors with multiple components:</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{c}x_1\\x_2\end{array}\right]
+
\left[\begin{array}{c}y_1\\y_2\end{array}\right]
=
\left[\begin{array}{c}y_1\\y_2\end{array}\right]
+
\left[\begin{array}{c}x_1\\x_2\end{array}\right]\text{.}
\end{equation*}
</div></article><article class="activity project-like" id="activity-20"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.1.1</span><span class="period">.</span>
</h6>
<p id="p-109">Consider each of the following properties of the real numbers \(\IR^1\text{.}\) Label each property as <em class="emphasis">valid</em> if the property also holds for two-dimensional Euclidean vectors \(\vec u,\vec v,\vec w\in\IR^2\) and scalars \(a,b\in\IR\text{,}\) and <em class="emphasis">invalid</em> if it does not.</p>
<ol class="decimal">
<li id="li-92"><p id="p-110">\(\vec u+(\vec v+\vec w)=
(\vec u+\vec v)+\vec w\text{.}\)</p></li>
<li id="li-93"><p id="p-111">\(\vec u+\vec v=
\vec v+\vec u\text{.}\)</p></li>
<li id="li-94"><p id="p-112">There exists some \(\vec z\) where \(\vec v+\vec z=\vec v\text{.}\)</p></li>
<li id="li-95"><p id="p-113">There exists some \(-\vec v\) where \(\vec v+(-\vec v)=\vec z\text{.}\)</p></li>
<li id="li-96"><p id="p-114">If \(\vec u\not=\vec v\text{,}\) then \(\frac{1}{2}(\vec u+\vec v)\) is the only vector equally distant from both \(\vec u\) and \(\vec v\)</p></li>
<li id="li-97"><p id="p-115">\(a(b\vec v)=(ab)\vec v\text{.}\)</p></li>
<li id="li-98"><p id="p-116">\(1\vec v=\vec v\text{.}\)</p></li>
<li id="li-99"><p id="p-117">If \(\vec u\not=\vec 0\text{,}\) then there exists some scalar \(c\) such that \(c\vec u=\vec v\text{.}\)</p></li>
<li id="li-100"><p id="p-118">\(a(\vec u+\vec v)=a\vec u+a\vec v\text{.}\)</p></li>
<li id="li-101"><p id="p-119">\((a+b)\vec v=a\vec v+b\vec v\text{.}\)</p></li>
</ol></article><article class="definition definition-like" id="definition-9"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">2.1.2</span><span class="period">.</span>
</h6>
<p id="p-120">A <dfn class="terminology">vector space</dfn> \(V\) is any collection of mathematical objects with associated addition \(\oplus\) and scalar multiplication \(\odot\) operations that satisfy the following properties. Let \(\vec u,\vec v,\vec w\) belong to \(V\text{,}\) and let \(a,b\) be scalar numbers.</p>
<ol class="decimal">
<li id="li-102"><p id="p-121">\(\vec u\oplus (\vec v\oplus \vec w)=
(\vec u\oplus \vec v)\oplus \vec w\text{.}\)</p></li>
<li id="li-103"><p id="p-122">\(\vec u\oplus \vec v=
\vec v\oplus \vec u\text{.}\)</p></li>
<li id="li-104"><p id="p-123">There exists some \(\vec z\) where \(\vec v\oplus \vec z=\vec v\text{.}\)</p></li>
<li id="li-105"><p id="p-124">There exists some \(-\vec v\) where \(\vec v\oplus (-\vec v)=\vec z\text{.}\)</p></li>
<li id="li-106"><p id="p-125">\(a(b\odot\vec v)=(ab)\odot\vec v\text{.}\)</p></li>
<li id="li-107"><p id="p-126">\(1\odot\vec v=\vec v\text{.}\)</p></li>
<li id="li-108"><p id="p-127">\(a\odot(\vec u\oplus \vec v)=(a\odot\vec u)\oplus(a\odot\vec v)\text{.}\)</p></li>
<li id="li-109"><p id="p-128">\((a+ b)\odot\vec v=(a\odot\vec v)\oplus(b\odot \vec v)\text{.}\)</p></li>
</ol></article><article class="observation remark-like" id="observation-4"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">2.1.3</span><span class="period">.</span>
</h6>
<p id="p-129">Every <dfn class="terminology">Euclidean vector space</dfn></p>
<div class="displaymath">
\begin{equation*}
\IR^n=\setBuilder{\left[\begin{array}{c}x_1\\x_2\\\vdots\\x_n\end{array}\right]}{x_1,x_2,\dots,x_n\in\IR}
\end{equation*}
</div>
<p data-braille="continuation">satisfies all eight requirements for the usual definitions of addition and scalar multiplication, but we will also study other types of vector spaces.</p></article><article class="observation remark-like" id="observation-5"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">2.1.4</span><span class="period">.</span>
</h6>
<p id="p-130">The space of \(m \times n\) <dfn class="terminology">matrices</dfn></p>
<div class="displaymath">
\begin{equation*}
M_{m,n}=\setBuilder{\left[\begin{array}{c}a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn} \end{array}\right]}
{a_{11},\ldots,a_{mn} \in\IR}
\end{equation*}
</div>
<p data-braille="continuation">satisfies all eight requirements for component-wise addition and scalar multiplication.</p></article><article class="remark remark-like" id="remark-8"><h6 class="heading">
<span class="type">Remark</span><span class="space"> </span><span class="codenumber">2.1.5</span><span class="period">.</span>
</h6>
<p id="p-131">Every Euclidean space \(\IR^n\) is a vector space, but there are other examples of vector spaces as well.</p>
<p id="p-132">For example, consider the set \(\IC\) of complex numbers with the usual defintions of addition and scalar multiplication, and let \(\vec u=a+b\mathbf{i}\text{,}\) \(\vec v=c+d\mathbf{i}\text{,}\) and \(\vec w=e+f\mathbf{i}\text{.}\) Then</p>
<div class="displaymath">
\begin{align*}
\vec u+(\vec v+\vec w)
&=
(a+b\mathbf{i})+((c+d\mathbf{i})+(e+f\mathbf{i}))\\
&=
(a+b\mathbf{i})+((c+e)+(d+f)\mathbf{i})
\\&=(a+c+e)+(b+d+f)\mathbf{i}
\\&=((a+c)+(b+d)\mathbf{i})+(e+f\mathbf{i})\\
&=
(\vec u+\vec v)+\vec w
\end{align*}
</div>
<p id="p-133">All eight properties can be verified in this way.</p></article><article class="remark remark-like" id="remark-9"><h6 class="heading">
<span class="type">Remark</span><span class="space"> </span><span class="codenumber">2.1.6</span><span class="period">.</span>
</h6>
<p id="p-134">The following sets are just a few examples of vector spaces, with the usual/natural operations for addition and scalar multiplication.</p>
<ul class="disc">
<li id="li-110"><p id="p-135">\(\IR^n\text{:}\) Euclidean vectors with \(n\) components.</p></li>
<li id="li-111"><p id="p-136">\(\IC\text{:}\) Complex numbers.</p></li>
<li id="li-112"><p id="p-137">\(M_{m,n}\text{:}\) Matrices of real numbers with \(m\) rows and \(n\) columns.</p></li>
<li id="li-113"><p id="p-138">\(\P^n\text{:}\) Polynomials of degree \(n\) or less.</p></li>
<li id="li-114"><p id="p-139">\(\P\text{:}\) Polynomials of any degree.</p></li>
<li id="li-115"><p id="p-140">\(C(\IR)\text{:}\) Real-valued continuous functions.</p></li>
</ul></article><article class="activity project-like" id="activity-21"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.1.2</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-7">
<p id="p-141">Consider the set \(V=\setBuilder{(x,y)}{y=e^x}\) with operations defined by</p>
<div class="displaymath">
\begin{equation*}
(x_1,y_1)\oplus (x_2,y_2)=(x_1+x_2,y_1y_2)
\hspace{3em}
c\odot (x_1,y_1)=(cx_1,y_1^c)\text{.}
\end{equation*}
</div>
</div>
<article class="task exercise-like" id="task-14"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-142">Show that \(V\) satisfies the distributive property</p>
<div class="displaymath">
\begin{equation*}
(a+b)\odot (x_1,y_1)=\left(a\odot (x_1,y_1)\right)\oplus \left(b\odot (x_1,y_1)\right)
\end{equation*}
</div>
<p data-braille="continuation">by simplifying both sides and verifying they are the same expression.</p></article><article class="task exercise-like" id="task-15"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-143">Show that \(V\) contains an additive identity element satisfying</p>
<div class="displaymath">
\begin{equation*}
(x_1,y_1)\oplus\vec{z}=(x_1,y_1)
\end{equation*}
</div>
<p data-braille="continuation">for all \((x_1,y_1)\in V\) by choosing appropriate values for \(\vec{z}=(\unknown,\unknown)\text{.}\)</p></article></article><article class="remark remark-like" id="remark-10"><h6 class="heading">
<span class="type">Remark</span><span class="space"> </span><span class="codenumber">2.1.7</span><span class="period">.</span>
</h6>
<p id="p-144">It turns out \(V=\setBuilder{(x,y)}{y=e^x}\) with operations defined by</p>
<div class="displaymath">
\begin{equation*}
(x_1,y_1)\oplus (x_2,y_2)=(x_1+x_2,y_1y_2)
\hspace{3em}
c\odot (x_1,y_1)=(cx_1,y_1^c)
\end{equation*}
</div>
<p data-braille="continuation">satisifes all eight properties.</p>
<p id="p-145">Thus, \(V\) is a vector space.</p></article><article class="activity project-like" id="activity-22"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.1.3</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-8">
<p id="p-146">Let \(V=\setBuilder{(x,y)}{x,y\in\IR}\) have operations defined by</p>
<div class="displaymath">
\begin{equation*}
(x_1,y_1)\oplus (x_2,y_2)=(x_1+y_1+x_2+y_2,x_1^2+x_2^2)
\hspace{3em}
c\odot (x_1,y_1)=(x_1^c,y_1+c-1)\text{.}
\end{equation*}
</div>
</div>
<article class="task exercise-like" id="task-16"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-147">Show that \(1\) is the scalar multiplication identity element by simplifying \(1\odot(x,y)\) to \((x,y)\text{.}\)</p></article><article class="task exercise-like" id="task-17"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-148">Show that \(V\) does not have an additive identity element by showing that \((0,-1)\oplus\vec z\not=(0,-1)\) no matter how \(\vec z=(z,w)\) is chosen.</p></article><article class="task exercise-like" id="task-18"><h6 class="heading"><span class="codenumber">(c)</span></h6>
<p id="p-149">Is \(V\) a vector space?</p></article></article><article class="activity project-like" id="activity-23"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.1.4</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-9">
<p id="p-150">Let \(V=\setBuilder{(x,y)}{x,y\in\IR}\) have operations defined by</p>
<div class="displaymath">
\begin{equation*}
(x_1,y_1)\oplus (x_2,y_2)=(x_1+x_2,y_1+3y_2)
\hspace{3em}
c\odot (x_1,y_1)=(cx_1,cy_1)
.
\end{equation*}
</div>
</div>
<article class="task exercise-like" id="task-19"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-151">Show that scalar multiplication distributes over vector addition, i.e.</p>
<div class="displaymath">
\begin{equation*}
c \odot \left( (x_1,y_1) \oplus (x_2,y_2) \right) = c\odot (x_1,y_1) \oplus c\odot (x_2,y_2)
\end{equation*}
</div>
<p data-braille="continuation">for <em class="emphasis">all</em> \(c\in \IR,\, (x_1,y_1),(x_2,y_2) \in V\text{.}\)</p></article><article class="task exercise-like" id="task-20"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-152">Show that vector addition is not associative, i.e.</p>
<div class="displaymath">
\begin{equation*}
(x_1,y_1) \oplus \left((x_2,y_2) \oplus (x_3,y_3)\right) \neq \left((x_1,y_1)\oplus (x_2,y_2)\right) \oplus (x_3,y_3)
\end{equation*}
</div>
<p data-braille="continuation">for <em class="emphasis">some</em> vectors \((x_1,y_1), (x_2,y_2), (x_3,y_3) \in V\text{.}\)</p></article><article class="task exercise-like" id="task-21"><h6 class="heading"><span class="codenumber">(c)</span></h6>
<p id="p-153">Is \(V\) a vector space?</p></article></article><section class="exercises" id="exercises-4"><h3 class="heading hide-type">
<span class="type">Exercises</span> <span class="codenumber">2.1.1</span> <span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercise-16"><h6 class="heading"><span class="codenumber">1<span class="period">.</span></span></h6>
<p id="p-154">Let \(V\) be the set of all pairs \((x,y)\) of real numbers together with the following operations:</p>
<div class="displaymath">
\begin{align*}
(x_1,y_1)\oplus (x_2,y_2)&= \left(4 \, x_{1} + x_{2},\,y_{1} + 3 \, y_{2}\right) \\
c \odot (x,y) &= \left(c x,\,c y\right) .
\end{align*}
</div>
<p id="p-155">(a) Show that scalar multiplication distributes over vector addition, that is:</p>
<div class="displaymath">
\begin{equation*}
c\odot \left((x_1,y_1)\oplus(x_2,y_2)\right)=c\odot(x_1,y_1)\oplus c\odot(x_2,y_2).
\end{equation*}
</div>
<p id="p-156">(b) Explain why \(V\) nonetheless is not a vector space.</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-16" id="answer-16"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-16"><div class="answer solution-like">
<p id="p-157">\(V\) is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:</p>
<ul class="disc">
<li id="li-116">vector addition is not associative</li>
<li id="li-117">vector addition is not commutative</li>
<li id="li-118">scalar multiplication does not distribute over scalar addition</li>
</ul>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-17"><h6 class="heading"><span class="codenumber">2<span class="period">.</span></span></h6>
<p id="p-158">Let \(V\) be the set of all pairs \((x,y)\) of real numbers together with the following operations:</p>
<div class="displaymath">
\begin{align*}
(x_1,y_1)\oplus (x_2,y_2)&= \left(x_{1} + x_{2},\,y_{1} + y_{2}\right) \\
c \odot (x,y) &= \left(c^{2} x,\,c^{3} y\right) .
\end{align*}
</div>
<p id="p-159">(a) Show that scalar multiplication distributes over vector addition, that is:</p>
<div class="displaymath">
\begin{equation*}
c\odot \left((x_1,y_1)\oplus(x_2,y_2)\right)=c\odot(x_1,y_1)\oplus c\odot(x_2,y_2).
\end{equation*}
</div>
<p id="p-160">(b) Explain why \(V\) nonetheless is not a vector space.</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-17" id="answer-17"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-17"><div class="answer solution-like">
<p id="p-161">\(V\) is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:</p>
<ul class="disc"><li id="li-119">scalar multiplication does not distribute over scalar addition</li></ul>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-18"><h6 class="heading"><span class="codenumber">3<span class="period">.</span></span></h6>
<p id="p-162">Let \(V\) be the set of all pairs \((x,y)\) of real numbers together with the following operations:</p>
<div class="displaymath">
\begin{align*}
(x_1,y_1)\oplus (x_2,y_2)&= \left(x_{1} + x_{2},\,y_{1} + y_{2}\right) \\
c \odot (x,y) &= \left(c x,\,c y - 3 \, c + 3\right) .
\end{align*}
</div>
<p id="p-163">(a) Show that scalar multiplication is associative, that is:</p>
<div class="displaymath">
\begin{equation*}
a\odot(b\odot (x,y))=(ab)\odot(x,y).
\end{equation*}
</div>
<p id="p-164">(b) Explain why \(V\) nonetheless is not a vector space.</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-18" id="answer-18"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-18"><div class="answer solution-like">
<p id="p-165">\(V\) is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:</p>
<ul class="disc">
<li id="li-120">scalar multiplication does not distribute over vector addition</li>
<li id="li-121">scalar multiplication does not distribute over scalar addition</li>
</ul>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-19"><h6 class="heading"><span class="codenumber">4<span class="period">.</span></span></h6>
<p id="p-166">Let \(V\) be the set of all pairs \((x,y)\) of real numbers together with the following operations:</p>
<div class="displaymath">
\begin{align*}
(x_1,y_1)\oplus (x_2,y_2)&= \left(x_{1} + x_{2},\,y_{1} + y_{2} - 5\right) \\
c \odot (x,y) &= \left(c x,\,c y\right) .
\end{align*}
</div>
<p id="p-167">(a) Show that vector addition is associative, that is:</p>
<div class="displaymath">
\begin{equation*}
\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).
\end{equation*}
</div>
<p id="p-168">(b) Explain why \(V\) nonetheless is not a vector space.</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-19" id="answer-19"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-19"><div class="answer solution-like">
<p id="p-169">\(V\) is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:</p>
<ul class="disc">
<li id="li-122">scalar multiplication does not distribute over vector addition</li>
<li id="li-123">scalar multiplication does not distribute over scalar addition</li>
</ul>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-20"><h6 class="heading"><span class="codenumber">5<span class="period">.</span></span></h6>
<p id="p-170">Let \(V\) be the set of all pairs \((x,y)\) of real numbers together with the following operations:</p>
<div class="displaymath">
\begin{align*}
(x_1,y_1)\oplus (x_2,y_2)&= \left(4 \, x_{1} + 4 \, x_{2},\,4 \, y_{1} + 4 \, y_{2}\right) \\
c \odot (x,y) &= \left(c x,\,c y\right) .
\end{align*}
</div>
<p id="p-171">(a) Show that scalar multiplication distributes over vector addition, that is:</p>
<div class="displaymath">
\begin{equation*}
c\odot \left((x_1,y_1)\oplus(x_2,y_2)\right)=c\odot(x_1,y_1)\oplus c\odot(x_2,y_2).
\end{equation*}
</div>
<p id="p-172">(b) Explain why \(V\) nonetheless is not a vector space.</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-20" id="answer-20"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-20"><div class="answer solution-like">
<p id="p-173">\(V\) is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:</p>
<ul class="disc">
<li id="li-124">vector addition is not associative</li>
<li id="li-125">scalar multiplication does not distribute over scalar addition</li>
</ul>
</div></div>
</div></article><p id="p-174"><em class="emphasis">Additional exercises available at <a class="external" href="https://checkit.clontz.org/banks/tbil-la/V1/" target="_blank">checkit.clontz.org</a>.</em></p></section></section></div></main>
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